51.1.50 problem 50

Internal problem ID [10320]
Book : First order enumerated odes
Section : section 1
Problem number : 50
Date solved : Tuesday, September 30, 2025 at 07:19:22 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} {y^{\prime }}^{2}&=x +y \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 22

ode:=diff(y(x),x)^2 = x+y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\left (\operatorname {LambertW}\left (-c_1 \,{\mathrm e}^{-\frac {x}{2}-1}\right )+1\right )}^{2}-x \]

✓ Mathematica. Time used: 9.77 (sec). Leaf size: 100

ode=(D[y[x],x])^2==x+y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to W\left (-e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right ){}^2+2 W\left (-e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right )-x+1\\ y(x)&\to W\left (e^{\frac {1}{2} (-x-2+c_1)}\right ){}^2+2 W\left (e^{\frac {1}{2} (-x-2+c_1)}\right )-x+1\\ y(x)&\to 1-x \end{align*}

✓ Sympy. Time used: 1.759 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x - y(x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ C_{1} + x + 2 \sqrt {x + y{\left (x \right )}} + 2 \log {\left (\sqrt {x + y{\left (x \right )}} - 1 \right )} = 0, \ C_{1} + x - 2 \sqrt {x + y{\left (x \right )}} + 2 \log {\left (\sqrt {x + y{\left (x \right )}} + 1 \right )} = 0\right ] \]